Consider a compact connected subset \(X\subset{\mathbb R}^{d}\) with nonempty interior, and let \(T:X\to X\) be a Borel measurable map. Assume that there exists a collection \((X_{i})_{i\in I}\) of finitely or countably many pairwise disjoint open sets satisfying that the closure of their union equals \(X\), such that for any \(i\in I\) the restriction of \(T\) to \(X_{i}\) is a \(C^{1}\)-diffeomorphism from \(X_{i}\) to \(T(X_{i})\) which can be extended to a \(C^{1}\)-map on the closure of \(X_{i}\). For any \(i\in I\) the restriction of \(T\) to \(X_{i}\) is called a branch of \(T\), and its inverse \(T_{i}:T(X_{i})\to X_{i}\) is called an inverse branch of \(T\). If there exists a norm \(\| .\| \) on \({\mathbb R}^{d}\) and a \(\beta >1\) with \(\| T(x)-T(y)\| \geq\| x-y\| \), then \(T\) is called expanding. The map \(T\) is called full branch map, if the closure of \(T(X_{i})\) equals \(X\) for every \(i\in I\). Finally, the map \(T\) is said to be real analytic, if there exists a bounded domain \(D\subset{\mathbb C}^{d}\) containing \(X\) such that \(T_{i}\) can be extended to a holomorphic function on \(D\) for every \(i\in I\).
In this paper the authors consider an expanding full branch real analytic map \(T\). Without loss of generality one may assume that \(I={\mathbb N}\), if \(I\) is countable, and \(I=\{1,2,\dots ,N\}\), if \(I\) is finite. Let \(\| .\| \) be a norm on \({\mathbb C}^{d\times d}\). If \(I\) is finite set \(\| T_{i}^{\prime}(z)\| =0\) for all \(i>N\). The map \(T\) is said to have uniformly summable derivatives, if \(\lim_{n\to\infty} \sup_{z\in D}\sum_{i=n}^{\infty} \| T_{i}^{\prime}(z)\| =0\). In the main result of this paper a map \(T\) with uniformly summable derivatives satisfying that the Lebesgue measure of \(X\setminus (\bigcup_{i\in I}X_{i})\) equals \(0\) is considered. It is proved that \(T\) has a unique absolutely continuous (with respect to Lebesgue measure) invariant Borel probability measure \(\mu\), the density function of \(\mu\) is real analytic and strictly positive on \(X\), and the dynamical system \((X,T,\mu )\) is exact.
Similar results have been obtained in the literature using different conditions, in particular the bounded distortion condition. The nice condition (uniformly summable derivatives) considered in this paper looks to be independent of the bounded distortion condition, and the authors announce a forth-coming paper where this independence is proved. Some other conditions known to imply the existence of an invariant absolutely continuous measure are discussed in this paper, and the authors give an example where these other conditions do not hold, but the results can be applied.
As the main technique in the proofs the transfer operator on suitable spaces of holomorphic spaces is used. One has to prove that the transfer operator is compact and positive. In fact, using a weighted version of the transfer operator a more general result is proved, replacing the Lebesgue measure by a finite nonsingular measure.